Inner mantles and iterated HOD

TL;DR

This paper introduces a class forcing notion that can produce models where the ground model is the η-th inner mantle and η-th iterated HOD, with controllable sequence lengths, answering a conjecture and demonstrating separability.

Contribution

It constructs a uniform class forcing notion that realizes prescribed lengths for inner mantle and iterated HOD sequences, resolving a conjecture and showing their independence.

Findings

Forces the ground model to be the η-th inner mantle of the extension.

Forces the ground model to be the η-th iterated HOD of the extension.

The lengths of inner mantle and iterated HOD sequences can be arbitrarily separated.

Abstract

We present a class forcing notion $\mathbb M(\eta)$, uniformly definable for ordinals $\eta$, which forces the ground model to be the $\eta$-th inner mantle of the extension, in which the sequence of inner mantles has length at least $\eta$. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that $\mathbb M(\eta)$ forces the ground model to be the $\eta$-th iterated HOD of the extension, where the sequence of iterated HODs has length at least $\eta$. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any two ordinals you please.